Note that the above conjecture does not say that \(\Theta \)  (the supremum of Wadge rank of universally Baire sets) is regular in \(V\)  assuming \(V\)  models \({\mathbf {MAX}(\mathsf {UB})}+(*)\text{-}\mathsf {UB}\)  (it could certainly have cofinality \(\omega _2\)  in \(V\) , and we conjecture it could not have cofinality
\(\omega _1\) ). In fact an argument of Woodin combined with the results of [1]} should bring that \(\Theta \)  cannot be regular in models of
\(\text{{\sf MM}}^{+++}\) .
